Monday, September 29, 2008

Aug-Sep Mystery MathAnagram Revealed - The Incomparable Poincare!

If nature were not beautiful, it would not be worth knowing, and if nature were not worth knowing, life would not be worth living. Poincare

The scientist does not study nature because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful. Poincare

Finally! Jules Henri Poincare!
Considering the stature of this brilliant individual and his contributions to mathematics, physics and philosophy, it's only fitting that he occupy our contest for at least two months (and i will leave his picture in the sidebar for awhile.). The standard reference for biographies will give you detailed background. Look here. For more of Poincare's profound quotes look here.

Our Winners Are:


Sean's Contribution:

(1) Henri Poincare
(2) He is described as a "polymath" and has been called "The Last Universalist" because he excelled at all the established mathematical fields of the time.
Also, Prince Louis-Victor de Broglie was awarded the Poincare Medal ( and the Prince of Monaco attended Poincares funeral (

Steve's Contribution:
1. Henri Poincare
2. Grigori Perelman has been credited with proving the Poincare Conjecture, although not without controversy. Perelman refused the Fields Medal offered to him after his proof was confirmed.
3. The clues that I used were in the Aug. 26 post, the word "relatively" and the reference to the "last of a dying breed." Poincare worked on relativity and is considered the last universalist. I also think the phrase "truly unique" refers to Poincare being a jewel; his first name is Jules.

If you're thinking that I feel reverence for Monsieur Poincare, you would be correct...

Friday, September 26, 2008

Geometry Investigation: Combining Similarity, Reflection, Paper Folding in an SAT-Type Problem

The problem/investigation below lends itself to a variety of approaches:
Similar triangles
Coordinate methods
Relationships Among ⊥ Bisector, Reflection, Paper Folding (Origami), and Symmetry

Note: Parts of the problem below are appropriate for middle schoolers.

Part I
Show that the length of the ⊥ bis segment EF is 7.5.
(1) Suggested Questions or Tips To Get Started:
(a) How does the length of EF compare to the diagonal? What would the rectangle look like if this length were equal to the diagonal?
(b) Mark (label) all known segments. Even if you cannot immediately determine a method for finding EF, what other segments can be more easily determined?
(c) Label congruent angles! This is critical!!

(2) Of course, the instructor doesn't have to give the answer away, but if the focus is on methods, you may want to consider this.

(3) Students usually struggle with recognizing similar triangles, particularly if they're not looking for them. You may want to give this as a hint after a few minutes. Even some strong honors students may be challenged. Whenever I make these kinds of claims, you can be assured I have already tried this kind of question and observed the reactions! However, coordinate methods are also powerful here and provide excellent review. It's more cumbersome than synthetic methods (Euclidean), but definitely worth discussing and is often a method of choice when other methods are not obvious. There are other methods as well using congruent right triangles, Pythagorean relationships and algebra. Then of course there are the wrong assumption methods: Assuming 3-4-5 triangles are the same as 30-60-90 triangles! As you circulate, see if anyone comes up with an answer involving √3!!

(4) For students preparing for SATs who have not had geometry for a year or more, this problem can be an excellent review even though the difficulty level is somewhat above the SATs. IMO, working a bit beyond the level of an assessment is usually the best way to prepare most students. Anyone agree?

(5) Problems that require thinking 'outside the box' and not mechanically are the most challenging for many students who view mathematics as algorithmically driven (procedures to follow).

Part II
Surely, group, you don't believe we will stop here! Generalize your result to determine an expression for the length of segment EF for an arbitrary rectangle whose dimensions are L and W. Your result will naturally be in terms of these two parameters (L and W).

Part III
Time to "reflect" on what you've done!
(a) Since EF is the ⊥ bis of diagonal AC, it follows that __ and __ are reflection images of each other.
(b) Take a large index card (ordinary filler paper is alright too but the stiffer the better). Fold the paper or card so that A and C coincide. What does this have to do with the original problem?
[Students need to "see" that the length of the fold or crease is the length of EF in the original problem!]
The mathematics of paper-folding (origami) is fascinating. The problem in this post is just an initial view of some of the underlying concepts. Students should have the opportunity to PLAY with the index card. Have them label all congruent segments and angles. Have them identify which segments or figures are reflections of each other. Have them look for a rhombus (explain why it is!), isosceles triangles, congruent right triangles, etc. Folding and unfolding the card is fascinating and instructive!

Note: Using geometry software is also instructive and should be considered as a supplement to the physical paper-folding here. There are also excellent sources for all this on the web. One of the best is the Math Forum at Drexel. An excellent investigation from the Forum can be found here. This is appropriate for both middle schoolers and high schoolers and is the kind of activity that is promoted on this blog. Check out the links at the bottom of this activity (some require subscription). It utilizes special software but can be modified. IMO, nothing replaces the need for students to hold an object in their hands.

Friday, September 19, 2008

"Fun" With Limits Early in Calculus

Update: The limit below can easily be derived using L'Hopital's theorem. The purpose of this article is to provide practice in algebra and limit manipulations, limit properties and the definition of the derivative prior to using this theorem.

Important Notes:

(1) The condition n ≠ 0 can be relaxed. At some point, students should be asked to analyze the need for restrictions.
(2) The instructor may well want to avoid giving students the above formula, preferring to have them derive it at least in the positive integer case (see comments below in red under "Developing the Problem").
(3) Note that m and n are not restricted to be positive integers. It is recommended that the instructor begin with this restriction on m and n to allow for an algebraic derivation.
(4) This is not an introductory limit exercise! Please read comments (red, bold) below about starting with concrete numerical values before attempting this generalization.

This is the time of year when Calculus students quickly move into those wonderful limit problems. Epsilon-delta arguments may not be as popular these days but the mechanics of limits are still the challenge for students. Those who have taught this know that students generally struggle with the algebraic simplifications and procedures. Other than these manipulations students generally feel this topic is easy:

Possible Student Thinking: "You just do some algebra, eliminate the "bad" factor in the denominator and plug in. Easy stuff!"

Naturally, if the assignments contain more theoretical limit problems, they may not feel that way!

On the other hand, the algebra can be a major stumbling block for the more challenging exercises. In this post, I will uncharacteristically deemphasize the theory behind the "cancel and plug in" technique and focus on the algebra at first. Then we will move on to relating the limit to the definition of the derivative and application of some important limit properties, in other words, theory! Of course, if L'Hopital's Theorem were introduced early on (in the chapter on differentiation), that would clearly be the method of choice for students!

If m and n in the limit above are positive integers, students can attempt to factor out "x-a" from the numerator and denominator and substitute x = a into the resulting reduced expression. However, many students struggle with such general factoring formulas (or may not have seen them.) Therefore, synthetic division can be used to generate the other factor. This reviews some nice Algebra 2 but what if m and n are not positive integers? What if they are rational or even irrational? Standard factoring techniques would not apply in general so what to do?

I certainly am not suggesting that the instructor begin with the general problem. In fact, I would 'concretize' the problem using a few special cases:
n=2,m=2 (This special case is worthwhile as it reviews basic definitions and limit properties).
n=4,m=5 (requires more sophisticated factoring or synthetic)

Based on these exercises, the instructor may ask students if they can develop a general formula for any positive integer exponents. this is in lieu of giving them the formula at the beginning.

After the definition of the derivative is given, students can attempt the more general version. This is a fairly sophisticated limit manipulation but one worth assigning. I may outline the method in an addendum to this post or in the comments or wait for one of our astute readers to contribute! As a hint, the technique I used is related to the derivation of L'Hopital's Theorem!

Monday, September 15, 2008

Reviewing Geometry for Class or SATs - Just a little tangent exercise?

The following problem is certainly appropriate for later in the year when geometry students reach this topic but it can also be used to review a considerable number of essential ideas in preparation for SATs, ACTs or just review in general. It's at the top end on the difficulty scale for these tests, but it's far from the AMC Contest!

Clarifications: Figures are not drawn to scale and the measure of ∠TAU is given in each diagram.

For each of the figures above, determine the following:
(a) the radius of each circle
(b) the length of minor arc TU in each circle

Have fun discovering a variety of approaches!

Variations? Generalizations? Choosing an angle other than special cases like 60 or 90 generally requires trig -- not that there's anything wrong with that!

Friday, September 12, 2008

The Largest Odd Factor of 90? Too Easy? How About A Million? A Googol!

Don't forget our MathAnagram for Aug-Sept. Thus far we have received a couple of correct responses. You are encouraged to make a conjecture!
Look here for directions. Here is the anagram again:


We would hope that by grade 5, most youngsters would be able to answer the first question in the title fairly rapidly and without a calculator. Or are you thinking many would incorrectly blurt out '9' as the answer?

Well, why should anyone care about finding the largest odd factor of some positive integer? Will it lead to a better understanding of the origins of the universe? Perhaps not, but these questions may deepen student understanding of

(a) Factors
(b) Prime factorization (particularly of powers of 10)
(c) The important concept that an even number may have odd factors but an odd number can never have any even factors!
(d) Other ideas...

Ninety is not a very large number so students will usually see that the answer is 45. However there are so many ways of looking at this simple result. So many important methods -- so little time! Further, a method that works effectively for 90 may not be as effective for 1,000,000 or a googol.

My suggestion is to give middle schoolers the 'million' problem, let them work with a partner, allow the use of a calculator and see what happens.

Here are some thoughts:

(a) Which of the following is more instructive, more important conceptually?

Writing 1,000,000 as 26⋅56, etc.,
Having students, on the calculator, divide 1,000,000 by 2, then the quotient by 2 and so on, until an odd result occurs

(b) Do these 2 approaches reinforce/develop the same concepts/skills or different ideas?

(c) Which method is most reasonable for 90? for 1,000,00? for a googol?

(d) Does the calculator enhance or not enhance understanding here? Does it depend on the number we start with?

Wednesday, September 10, 2008

This Logic Challenge is 'Par for the Course'!

Don't forget our MathAnagram for Aug-Sept. Thus far we have received a couple of correct responses. You are encouraged to make a conjecture!
Look here for directions. Here is the anagram again:


A former student sent me a wonderful reasoning problem involving mean, median, and mode, so it is accessible to middle schoolers. The question came from his teacher so I decided to revise it, put it in a different context but preserve the essence of the logic. The student will need to know some basics of scoring in golf but most of it should be clear. If not,
this may help.

This kind of question will frustrate some but reasonable frustration can often lead to 'pearls of wisdom.' Clear thinking and careful attention to detail is necessary. Certainly basic knowledge of measures of central tendency is a prerequisite, but this question can also serve to review these ideas.

Have fun with it yourself and, if you can, try it as a 5-minute warmup in class, preferably with students working in pairs. Let us know if they make a 'hole in one'! Again, thanks to my student and his teacher for the original source of this challenge.

Alex played 18 holes of golf and we know the following information:
His maximum score on any hole was a '5' and he shot this on six holes.
His median score on the 18 holes was 4.
The mode was 3.
What was the lowest possible mean score he could have achieved on the 18 holes?

Express your answer to 2 places (rounded).

Sunday, September 7, 2008

Remainders and Number Theory Challenges for Middle School and Beyond

Edit: #4 below has been corrected. I am indebted to one of mathmom's astute students for catching my error!

Number theory is part of many states' standards but usually only at a basic level (factors, multiples, primes, composites, gcf, lcm). Below you will find a problem for your students to work on (preferably with partner). It is not an introductory problem using remainders so they would have needed to do preliminary work beforehand.

Here are some suggestions for developing the foundation for today's challenge problem:

(1 ) List the first 5 positive integers which leave a remainder of 1 when divided by 2? Describe, in general, such positive integers.

(2) List the first 5 positive integers which leave a remainder of 3 when divided by 13? If you subtract 3 from each of these, what do you notice? Explain!

(3) List the first 5 positive integers which leave a remainder of 12 when divided by 13. If you subtract 12 from each of these, what do you notice? If, instead you ADD 1 to each of the 5 positive integers, what do you notice? Explain!

(4) What is the least positive integer N, greater than 1, which leaves a remainder of 1 when divided by 2, 3, 4 or 5? [Ans: 61]
Note: The word 'or' may be confusing or inaccurate here. Modify as needed!

Now for today's challenge (allow use of calculator):

What is the least positive integer which satisfies ALL of the following:
leaves a remainder of 1 when divided by 2
leaves a remainder of 2 when divided by 3
leaves a remainder of 3 when divided by 4
leaves a remainder of 4 when divided by 5
leaves a remainder of 5 when divided by 6
leaves a remainder of 6 when divided by 7
leaves a remainder of 7 when divided by 8
leaves a remainder of 8 when divided by 9.

This challenge looks harder than it is. Variations of these often appear on math contests for middle school and beyond. Simpler versions like example (4) above have appeared on the SATs.

Of course, modular arithmetic and congruences would make this problem trivial but that is non-standard and requires more time to develop.

I will not yet post the answer or possible solution...

Thursday, September 4, 2008

Achieve/ADP Algebra 2 End of Course Exam Report/Findings and MathNotations Commentary - Part I

Addendum: This commentary will shortly be followed by Part II which will focus on some of the following issues:
(a) Why does Achieve stress that the content is Advanced Algebra when it appears to be primarily standard Algebra II.
(b) Significant discrepancy in student performance between multiple choice vs. student-constructed and open-ended questions; implications for other standardized tests (do students do better or worse on student-constructed questions on SATs?)
(c) Do the results on this test suggest that Algebra 1 should have been the first such "standardized" test? In other words is the real issue here weaknesses in Algebra 1 background?

Note: Any facts or figures cited below come from the recently released report from Achieve. You will find a link to the full report below. For further background on the exam and links to released questions, link to my post from April 15, 2008.

If your school district participated this past May or June in the first administration of the Algebra 2 End of Course Exam developed by Pearson for the American Diploma Project you already know the results have been published. Nearly 90,000 students from 12 of the 14 states in the ADP partnership participated.

This post will provide an overview of the full report and some commentary. For general information regarding the exam, look here. Click on the next to last link in the right sidebar - it will take you to a new page which provides an overview of the Annual Report for this exam. The first link will give you the full pdf report. If you're familiar with the Exam, go directly to this new page. Also, for an excellent overview and objective commentary, the Achieve group obtained permission to link to the article in a recent Education Week (3rd link down on the report page). You must adhere to the restrictions about reproduction of this article but it's well worth reading.

When the Calculus Reform group wanted to impact curriculum and instruction in high school (and undergraduate) calculus, how did they do it? They knew the key was to change the AP Calculus Exam: the format, the content, the emphasis (less mechanics, more conceptual, more data-based/modeling open-ended questions, more use of graphing calculator technology).

If NCTM's reforms have not fully been felt K-12 (particularly 7-12), perhaps it's because there is no standardized assessment out there that truly reflects these reforms. It's true that some standardized tests now reflect more problem-solving, data analysis and conceptual understanding, but there's no single powerful test for grades 6-7-8 that will drive change in the classroom. Each individual state has its own independently developed and scored assessment for each grade level now, but the content, difficulty and quality of these tests vary widely. This is why I felt the benefits from the Achieve program far outweighed the potential risks.

Predictably, each time there is a significant change in the AP Exams or the SATs, scores initially drop. This is to be expected and desirable since the appropriate response to this is to understand what needs to be changed in content and instruction. All of the reports and recommendations from the most esteemed mathematics groups/panels have had little effect compared to the more immediate results that follow a drop in scores on some standardized test.

I read the report thoroughly. Passing scores or cutoffs were not determined at this point. Average raw scores and percents were reported for each grade level. It is very hard to draw informed conclusions without an analysis of the questions themselves since the level of difficulty, content and format of these questions are critical factors in performance. Further, scores on a first administration of any standardized test are expected to be lower.

I have not received permission from Achieve to reproduce excerpts so I will summarize major findings.

First I will provide some additional background on the format of the exam itself that you will need to make sense of the results below:

Three types of questions: Multiple-Choice, Short Answer and Extended Response.
A total of 76 raw score points, broken down as follows:

Multiple Choice: 46 questions - 1 pt. ea.
Short Answer: 7 questions - 2 pts. ea.
Extended Response: 4 questions - 4 pts. ea.

Further, the questions are broken into 3 cognitive levels with the majority of questions at Level 2 which "requires students to make some decisions as to how to approaqch the problem or activity."

There was a calculator and a non-calculator part.

For more info regarding the actual topics tested, refer to my link in the first paragraph of this post.

Based on a max of 76 raw score points, the average number of points scored ranged from a high of 39 points (about 50%) for 8th graders to a low of 16 points (about 20%) for 12th graders with a fairly steady decline from 8th through 12th.

MathNotation Commentary:
The decrease from 8th to 12th is easy to explain as the more capable students take the course earlier in accelerated classes. The 8th grade population was of course a very small sample but you get the idea. More significant is the average 24% correct for grade 11, the most common grade for students to take this course (in fact, the number of juniors nearly equaled all of the other grades combined). I'm not surprised by this low percentage for several reasons:
(a) First administration of the test
(b) We already knew there was an issue here or there would have been no impetus for developing uniform standards and a standardized assessment. Are these results so dramatically different from the TIMSS findings? I don't think so. However, there is no cause for alarm. The appropriate response is to provide the data to the states and local districts so that deficiencies can be addressed. I'm not at all concerned about the "Now they'll start teaching to the test" critiques. Those arguments were leveled at AP teachers as well. However, good assessments drive change in content and instruction. Excellent tests can enhance learning -- that's all I ever care about. If this Algebra 2 exam leads to more consistency and higher quality of curriculum and instruction, then everyone should be elated. Unfortunately, each side in the Math Wars will spin the results to make a case for their position. Similarly, Achieve, individual states (governors, state ed departments) will put their spin on it as well. It's up to the reader to become as highly informed as possible to draw her/his own conclusions. Overall, I'm not surprised by the initial outcome.

To be continued...

Tuesday, September 2, 2008

Setting the Tone in Precalculus - Another Coordinate Investigation

Note: Read the first comment I posted which suggests a purely Euclidean geometry approach to this problem...

Don't forget our MathAnagram for Aug-Sept. Thus far we have received a couple of correct responses. You are encouraged to make a conjecture!
Look here for directions. Here is the anagram again:


Tangent problems are usually the domain of calculus but we can keep them within the reach of geometry and algebra if we restrict our attention to circles. The calculus student spends a considerable amount of time solving a wide variety of "tangent to the curve" exercises. As any calculus instructor will tell you, many of the harder problems ask students to determine the equations of the tangent to a curve from a point not on the curve. The issue there is not the calculus. It's all about an understanding of the interface between the algebra and geometry, the essence of coordinate methods. I developed this investigation specifically to address this issue before students enter calculus. Might be another "fun" problem to start the year off with. If nothing else, it will establish the rigor of your precalculus course early on!

Part I of Investigation
Determine the coordinates of the points of tangency for the tangent lines to the unit circle from the point (0,2).

Note: Unit circle refers to the circle of radius 1, center (0,0).

The remaining parts will all refer to this same circle.

Part II of Investigation
Repeat part I for (0,3) and (0,4).
Write your observations, conjectures.

Part III of Investigation
Show that the y-coordinate of the points of tangency for the tangent lines to the unit circle from the point (0,k) is 1/k, where k ≥ 1.

Notes, Comments...
(1) The result of Part III suggests that as k increases, the y-coordinate of the point of tangency decreases (inverse ratio). Ask students what happens as k approaches 1.
Students should make sense of this visually by sketching tangent lines from various points on the y-axis above the circle.
(2) There are several effective methods for solving the above parts, however, one needs to know the fundamental relationship between a tangent line and the radius drawn to the point of tangency. From that point on, one can represent the slope in two ways or represent the y-coordinate of the point of tangent in two ways. This requires strong understanding of coordinates, graphs and algebraic relationships. You may find other methods -- share them! BTW, one could also use trig methods.
(3) I chose the unit circle and a point on the y-axis for simplicity so that the student could focus on essential ideas. However, one could generalize the result to any circle and any point outside. Have fun with that!
(4) Anyone mildly surprised by the reciprocal relationship between the y-coordinate of the point on the y-axis and the y-coordinate of the point of tangency? Can anyone make sense of that?